A Counterexample to the Gaussian Completely Monotone Conjecture

TL;DR

This paper presents a specific probability measure that disproves the Gaussian completely monotone conjecture by showing the fifth derivative of entropy can be positive, challenging several longstanding conjectures.

Contribution

It provides the first explicit counterexample to the GCM conjecture, also impacting related conjectures on Gaussian optimality and entropy power.

Findings

Disproved the GCM conjecture with an explicit measure.

Showed the fifth derivative of entropy can be positive.

Implication for the existence of log-concave measures violating GCM.

Abstract

We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.